Snapping vertex with edge slide but cannot be parallel to snapped vertex
In the general case where you're looking for closest point to an arbitrary vertex from an arbitrarily rotated edge I'm not aware of a faster method, and you may have to use TinyCad to do some edge intersection projection - but, that said, here is my best attempt to make it quicker:
Starting from this point and attempting to make the highlighted vertex slide along its edge and intersect with the notional plane extending along the Y axis from the floating edge behind it:

Extrude a temporary face from the rear edge until it intersects the target edge:

Set snapping options to Face snap:

Perform a vertex slide on the target vertex and snap to the face:

Delete the temporary construction geometry, and confirm that the vertex is now in the intended position:
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Obviously this is several steps, so it may take quite a while for many vertices, but if all of the snap target is a single section you could probably extrude it all at once, and possibly then separate the new geometry to a separate object to make it easier to delete later.  Then it's just a matter of selecting each vertex or group of vertices in turn.  Vertex slide will work on multiple, but snap will typically fail on groups and does in this example, so you need to select and snap move them individually.